On the asymptotic behaviour of the number of subgroups of finite abelian groups
type: Item
instance of: scholarly article
title: On the asymptotic behaviour of the number of subgroups of finite abelian groups
zbMATH Open document ID: 0907.11030
author: Gautami Bhowmik
author: Jie Wu
published in: Archiv der Mathematik
publication date: 1998-07-27T00:00:00Z
review text: Let \[ t_2(n) = \sum_{| {\mathcal G}| =n,\;r({\mathcal G})\leq 2}\tau({\mathcal G}), \] where \(\tau({\mathcal G})\) denotes the number of subgroups of a finite abelian group \({\mathcal G}, r({\mathcal G})\) is the rank of \({\mathcal G}\), and \(| {\mathcal G}| \) is the order of \({\mathcal G}\). The group \({\mathcal G}\) has rank \(r\) if \[ {\mathcal G} \cong \mathbb Z/n_1\mathbb Z \otimes \cdots \otimes \mathbb Z/n_r\mathbb Z, \] where \(n_j\mid n_{j+1}\) for \(j = 1,\cdots,r - 1\). We set \[ T(x) = \sum_{n\leq x}t_2(n) = \sum_{| {\mathcal G}| \leq x,r({\mathcal G})\leq 2}\tau({\mathcal G}), \] so that one has \[ T(x) = K_1x\log^2x + K_2x\log x + K_3x + \Delta(x), \] where \(K_j\) are effective constants and \(\Delta(x)\) is to be considered as the error term in the asymptotic formula for \(T(x)\). The bound \(\Delta(x) \ll x^{31/43+\varepsilon}\) was proved by \textit{G. Bhowmik} and \textit{H. Menzer} [Abh. Math. Semin. Univ. Hamb. 67, 117-121 (1997; Zbl 0890.11027)]. Further \textit{H. Menzer} [Proc. Conf. analytic number theory (ed. W. G. Nowak), Vienna 1996, Universität Wien, 181-188 (1996; Zbl 0879.11049)] improved the above bound to \(\Delta(x)\ll x^{9/14+\varepsilon}\) by using two estimates in the three-dimensional asymmetric divisor problem. In the present work the authors obtain the bound \(\Delta(x)\ll x^{5/8}\log^4x\). This is achieved by obtaining the same bound, by the intricate technique of multiple exponential sums, for the error term in the asymptotic formula for the summatory function of \(f(n) := \sum_{n=n_1n_2n_3^2}n_3\), where \(n_1,n_2,n_3\) are natural numbers. This result, which is hitherto the sharpest one, is certainly of independent interest. From the shape of the relevant generating function it seems reasonable to expect that, for suitable constants \(A_j\), one has \[ \Delta(x) = A_1x^{1/2}\log^2x + A_2x^{1/2}\log x + A_3x^{1/2} + E(x), \] where \(E(x)\) may be considered as a further error term. The authors prove that, for any g…
Mathematics Subject Classification ID: 11N45
Mathematics Subject Classification ID: 20K27
Mathematics Subject Classification ID: 11L07
Mathematics Subject Classification ID: 20K01
zbMATH DE Number: 1145400
zbMATH Keywords: abelian groups
zbMATH Keywords: rank
zbMATH Keywords: subgroups
zbMATH Keywords: Smith normal form
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: Dirichlet hyperbola principle
zbMATH Keywords: exponential sums
zbMATH Keywords: asymptotic formula
zbMATH Keywords: multiple exponential sums
zbMATH Keywords: number of distinct prime divisors
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
full work available at URL: s000130050098
OpenAlex ID: W2017535590
DOI: 10.1007/S000130050098
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description: scientific article; zbMATH DE number 1145400